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User blog:Emlightened/A Largest Number
This is an essay on largest numbers. Go figure. There's a remark at the end you should check out if the rest isn't all that interesting. First, a new largest number (a yet larger one is defined at the bottom of the page). Work in ZFC+WA0, the theory of ZFC with an elementary embedding symbol \(j:V \to V\) without replacement for formulae involving \(j\) and only bounded seperation for formulae involving \(j\). The critical point is denoted \(\kappa_0\), and let \(\kappa_{n+1} = j(\kappa_n)\). The language used is \(\langle \in, j, V_{\kappa_0} \rangle\). *k-Wholeness Number: The largest integer uniquely defined by a \(\Delta_0\) formula with quantifier depth \(\leq k\) and \(\leq 2^k\) occurances of the symbols \(j\), \(V_{\kappa_0}\). *k-Large Wholeness Number: The largest integer uniquely defined by a formula with quantifier depth \(\leq k\) and \(\leq 2^k\) occurances of the symbols \(j\), \(V_{\kappa_0}\). The difference is that LWN allows arbitrary (small enough) formulae, and WN only allows \(\Delta_0\) formulae. Now, this is a new largest number. Other largest numbers have typically considered all formulae over \(V\) (equivalently here \(V_{\kappa_0}\)), or extended \(V\) in some (hopefully unique, consistency-preserving) way that is still an element of \(V_{\alpha}\) for some \(\alpha \ll j(\kappa_0)\) and probably doesn't even contain all of \(V_{\kappa_0+1}\). All of the defined structures that extend \(V\) do so in a way that can be done internally using the elementary embedding, and are therefore smaller. Okay then, now what about if we just add \(j\) (\(V_{\kappa_0}\) is easily definable if our formulae aren't restricted to \(\Delta_0\)) to these extensions? New largest number, right? Well, that would work if you consider LWN to be the largest number defined, certainly. The language is still more expressive than with an I3 rank-into-rank but not the Wholeness Axiom (and hence it's not just a matter of large cardinal strength; the above argument applies with rank-into-ranks too). But is it largest? One reason you may disagree is that you contest the consistency of WA, which is considerable (greater than supercompacts, extensibles, Vopenka's principle, huge cardinals etc.). Another is that you contest a principle of ZFC, such as excluded middle, powerset, unbounded separation/collection or choice, and either think the above is nonsense (because it relies on something that doesn't make sense) or weak (because removing one of these principles allows you to make something stronger, such as (possibly) ZF + Reinhardt Cardinal). However, I'll argue that there's a perspective where LWN is not well-defined but WN is and that moreover from this perspective the previous methods for making new largest numbers are essentially trivial extensions. Often, we want to speak of 'all things' in mathematics, such as all groups or all sets, and this has proper-class size. When it does, we can only do it to a limited (first-order) extent, because of the size limitations of the universe. Now, a natural way to circumvent this is with multiple universes. It is sufficient to have each universe contained in a larger one; there is no reason for replacement to act on universes, so a \(\omega\)-high stack is sufficient. The most well-known ways of doing this are correct cardinals and Grothendick universes. The latter of these, however, has a major deficiency: what is true in one universe may not be true in another (such as "there is a largest inaccessible cardinal"). Correct cardinals are only slightly better: they agree on first-order formulae, but not generally on higher-order formulae. What about if we wanted second-order formulae? This, as it happens, require the two universes \(V_\kappa\) and \(\V_\lambda\) to have an elementary embedding \(j':V_{\kappa+1}\to V_{\lambda+1}\). And, because we want them to agree on all higher-order formulae (not just \(\omega\)-order), this leads to a \(j':V_{\lambda}\to V_{\lambda'}\) with critical point \(\kappa\), and more generally for the universes to all be \(n\)-huge (each natural \(n\)) and then to the Wholeness Axiom, with universes \(V_{\kappa_n}\). However, this only gives us bounded separation and replacement in \(V_{\kappa_\omega}\) (what we axiomise), as they are only unbounded in the individual universes. By elementarity, however, we retrieve full separation and replacement for formulae not involving \(j\), and we also have bounded separation for formulae involving \(j\). In fact, we cannot justify any stronger separation, as this would be unbounded in \(V_{\kappa_\omega}\), which isn't even a universe (and fails to have the same second-order properties as the actual universes). In particular, this does not justify LWN as valid but does justify WN. Now, what's to stop us from simply adding \(j\) and \(V_{\kappa_0}\) to the language we work in, and then invoking one of the previously defined constructions for a largest number? Well, any construction is generally done relative to \(V\), which here is \(V_{\kappa_n}\) (any \(n\)), so the construction can just be done internally to \(V_\lambda\) for some \(\lambda \ll \kappa_{n+1}\). One possible avenue of escape is that in our final definition, we only have finite quantifier depth. Perhaps we could define our formulae by transfinite induction? Yet if we did that for \(V_{\kappa_n}\) (\(n\) the largest we could make), we could do it internally for a larger \(n\). I make the following conjecture: *If \(i-N\) is a family of numbers indexed by \(i\) in a system that extends and is subject to the restrictions of the WN then there is a definable function \(f\) such that \(f(k)-WN > k-N\). Note that (k-WN)-WN isn't well defined, as this would require induction on universes (up to k-WN), which is disallowed, and that this essentially means that the Wholeness Numbers are the biggest definable numbers. Remark: Part of the point of this post is to point out that the idea that there is a single largest (defined) number/fastest growing function is fictitious. Regardless of if the definitions are valid, you've got to consider what someone considers valid. For instance, you might take the above stance, of ZFC+WA0 being the foundation, or you might be of a weaker strain, or you might be an intuitionist, or finitist, or predicitavist, or constructivist, or only believe in DC but not AC, or believe the computability axiom, or consider the limited principle of omniscience to be the only way to construct uncomputable functions, or not believe ZFC is consistent (and using correct carinals implies that), or use a foundations with proper classes (which would make some largest numbers weak, and drastically increase teh consistency strength of others), or build the universe up in a well-founded way which is predicative on sets but not formulae, or whatever. And, yeah, ZFC is the main current foundation of mathematics (with HoTT likely to soon become a popular competitor). But whenever a largest number is defined, the principles which it is declared largest under are also defined, and these are not universal. There may even be a 'computable' fragment of \(L_{\omega_1}\) containing only computable things based on a maximal type theory, although I've seen nothing in the literature hinting at this. Such a theory would actually be where part of googology lives, as everything is built in a well-founded computable manner, and you can define functions according to whatever well-founded induction principle takes your fancy, but you can't just diagonalise over an arbitrary consistent theory. Second-to-lastly, I don't agree that the theory presented is correct. I prefer to work in weaker settings, such as constructively, and without power sets ever being completed infinities (like they are in second-order arithmetic). Oh, lastly, all of the post except for the remark was made a couple of weeks ago, so I'm sorry if I say something wrong in the comments. Category:Blog posts